logical relations and contextual equivalence

I am looking for (in)completeness results of logical relations
w.r.t. contextual equivalence---that is, whether contextually
equivalent terms can always be logically related---in typed
lambda-calculi.
In particular, I am interested in such results in the following
settings:
- with universal types or existential types
- with divergence or without divergence
- under call-by-name or call-by-value evaluation
My understanding is that logical relations are
- complete in a call-by-name setting with divergence and universal types
- incomplete in call-by-value settings with existential types
- incomplete in call-by-value settings with universal types
but corrections and any other information are _very_ welcome.
Thank you in advance,
// Eijiro Sumii (http://www.yl.is.s.u-tokyo.ac.jp/~sumii/)
//
// Ph.D. Student at Department of IS, University of Tokyo
// Visiting Scholar at Department of CIS, University of Pennsylvania